Ocean models are used to predict currents, temperature, and salinity of the earth's oceans, much like weather forecasting. As supercomputer hardware changes with evolving technology, models must be updated, and sometimes rewritten. Here we document Omega, a new ocean model that was designed to run on the world's fastest supercomputers. Testing shows that Omega accurately solves the model equations, and runs efficiently on many different computer architectures, including exascale computers.